Hillary Clinton will lavish praise on Barack Obama today when she formally concedes the race for the Democratic nomination at a rally for her supporters after the two held a secret meeting in Washington.

Her campaign has chosen a grand venue in Washington for today's event, allowing her to make a farewell to supporters as well as a formal endorsement of Obama.

The endorsement follows the first direct meeting between Obama and Clinton for months after an often acrimonious campaign, at the home of a Democratic senator, Dianne Feinstein. The candidates arrived alone for their late-night meeting - without spouses or staff - and met for about an hour.

Feinstein said yesterday she left them alone in her living room while she went upstairs. When she came down, she said they were laughing and seemed to be getting along well. "It was a good first step," she told reporters. She said she did not hear any shouting from upstairs.

They went to extraordinary lengths to hold the meeting out of sight of the media. Obama, in a diversionary tactic, sent his travelling press corps to Chicago while he remained in Washington.

CNN and other news stations spent hours showing live footage from outside Clinton's Washington home, with commentators saying Obama had made a significant compromise in going there, only to discover they were meeting at Feinstein's home.

The first sign the press had that something was afoot was when Obama failed to appear at Washington's Dulles airport.

Robert Gibbs, his communications director, told the press that there had been a change of plan and Obama would not be accompanying them.

A television reporter managed to send a text message to her office as the engines started and the Washington press corps fanned out across the city hunting for the two candidates. The favoured location was the Clintons' home near the British embassy on Massachusetts Avenue, but others headed for Obama's flat on Capitol Hill, as well as the homes of the candidates' friends.

When the travelling press corps landed in Chicago, Gibbs told them there was a meeting but gave no further information. Calls to other press aides resulted in silence, in part because they did not know where the candidates were either.

Clinton made the first move, phoning Obama on Thursday afternoon suggesting they meet. Obama had been campaigning throughout the day in Virginia and had been looking forward to returning home to Chicago for a three-day break.

After the meeting, Obama and Clinton put out a short joint press statement saying it had been a productive meeting about working for victory in November.

It was the first sign of cooperation between the camps for months. In an indication of the depth of bad feeling between the two camps, Congressman Rob Andrews, a Clinton supporter claimed a senior member of her team had discussed with him exploiting tensions between Jews and African-Americans.

"There have been signals coming out of the Clinton campaign that have racial overtones that indeed disturb me," Andrews told the New Jersey Star-Ledger. "Frankly, I had a private conversation with a high-ranking person in the campaign ... that used a racial line of argument that I found very disconcerting. It was extremely disconcerting given the rank of this person. It was very disturbing."

A Clinton campaign official denied any attempt to stir such tensions.

The two are looking at how to unite the Democratic party after a 16-month campaign marked by bitter exchanges. The party split along lines of race, age, class and gender.

Obama won the overwhelming support of African-Americans, young people and professionals while Clinton took a majority of women, Hispanics and white, working-class males.

Clinton was on the receiving end of a party backlash all week after failing to congratulate Obama when he secured the majority needed for the nomination Tuesday. Aides said yesterday she would remedy that today. She will also speak about the difficulties of running as a woman and about the issues on which she will continue to fight, such as the introduction of universal healthcare.

"She wants to do everything she can to bring the party together," Feinstein told reporters. "She wants to do everything she can to see the people who voted for her have their voices heard ... And she wants to have a working relationship with Senator Obama, and I think it's a very positive thing."

One of her leading supporters, fellow New York senator Charles Schumer, said yesterday she would accept the vice-presidential slot if offered it. "She has said if Senator Obama should want her to be vice-president ... she will accept that. But on the other hand if he chooses someone else she will work just as hard for the party in November," he told ABC television.

But hopes among her supporters that Obama would offer her the slot were receding yesterday. One aide speculated that a cabinet post was also unlikely, saying she would be better off in the Senate.

John Edwards, a contender for the Democratic nomination until he quit in January, ruled himself out as a vice-presidential candidate in interviews published in Spanish papers yesterday. He said he had been the vice-presidential candidate in 2004 and did not want to repeat the experience.

One of the topics high on Clinton's agenda is seeking Obama's help in paying off the $30m (£15m) in campaign debts she accumulated.

· This article was amended on Tuesday June 10 2008. The member of the US House of Representatives referred to above is Rob Andrews, not Rob Edwards. This has been corrected.